Contact (mechanics)
The contact mechanics deals with the calculation of elastic, viscoelastic or plastic bodies in static or dynamic contact. Contact mechanics is a fundamental engineering science discipline, which is essential for a safe and energy-saving design of technical systems.
It is of interest for applications, such as wheel -rail contact, clutches, brakes, tires, lubricants and ball bearings, internal combustion engines, joints, seals, forming, machining, ultrasonic welding, electrical contacts and many others. Their tasks range from the strength analysis of contact and connection elements on the influence of friction and wear by lubrication or material design to applications in micro -and nano- technology.
History
The classical contact mechanics is mainly associated with Heinrich Hertz. In 1882, Hertz solved the problem of the contact between two elastic bodies with curved surfaces ( see the article Hertzian pressure ). This classic result is even today a basis of contact mechanics. Other early analytical work on this subject go back to JV Boussinesq and Cerruti V..
It was not until almost a century later found Johnson, Kendall and Roberts a similar solution as that of Hertz for an adhesive contact ( JKR theory).
A further advance our knowledge of contact mechanics is in the middle of the 20th century and is connected with the name of Bowden and Tabor. They have pointed to the importance of the first roughness of the contacting bodies. Due to the roughness of the true contact area between the friction partners is typically orders of magnitude smaller than the apparent area. This insight suddenly changed the direction also of many tribological investigations. The work of Bowden and Tabor have initiated a number of theories of contact mechanics of rough surfaces.
A pioneer work in this field, especially the work of Archard (1957 ) should be mentioned, who has come to the conclusion that even in the contact of elastic, rough surfaces, the contact surface is approximately proportional to the normal force. Other important contributions are connected with the name of Greenwood and Williamson (1966 ), Bush ( 1975) and Persson ( 2002). The main result of this work is that the true contact area for rough surfaces is roughly proportional to the normal force, while the conditions (pressure, size of the micro- contact) only weakly depend in individual micro-contacts on the load.
Today, many tasks of the contact mechanics are processed with simulation programs based on the finite element method or the boundary element method. For this there are a large number of scientific papers, some are found in addition to the basics of numerical contact mechanics in the books by Laursen (2002) and Wriggers (2006).
Classical Contact Tasks
Contact between a sphere and an elastic half-space
If an elastic sphere of radius into an elastic half-space indented by the amount ( depth of indentation ), so a contact area with the radius forms. The force required is equal to
,
In which
.
And here are the elastic moduli and Poisson's ratios and the two bodies.
Contact between two elastic spheres
If two spheres with the radii and in contact, these equations are still valid with the radius according to
The pressure distribution in the contact area is given by
With
.
The maximum shear stress lies in the interior, for at.
Contact between two crossed cylinders of equal radii
Is equivalent to the contact between a sphere having a radius and a plane ( see above).
Contact between a rigid cylinder and an elastic half-space
If a rigid cylindrical stamp pressed with the radius into an elastic half-space, the pressure distribution is by
Given with
.
The relationship between the penetration depth and the normal force is
.
Contact between a rigid conical indenter and the elastic half-space
In Indentierung an elastic half-space by a rigid conical indenter, the indentation depth and the contact radius by the relation
Given. is the angle between the plane and the side surface of the cone. The pressure distribution in the form of
.
The tension has been at the apex of the cone ( in the center of the contact area ) a logarithmic singularity. The total force is calculated as
.
Contact between two cylinders with parallel axes
In the event of a contact between two cylinders with parallel axes, the force is linearly proportional to the indentation depth:
.
The radius of curvature does not appear in this respect. The half contact width is the same relationship
,
With
Where, as in the contact between two spheres. The maximum pressure is equal to
.
Contact between rough surfaces
When two bodies are pressed against each other with rough surfaces, the real contact area is initially much less than the apparent surface. For a contact between a " random rough " surface and an elastic half-space, the real contact area is proportional to the normal force and is given by the equation
Optionally, wherein the root mean square slope of the surface and. The mean pressure in the true contact area
Is calculated to a good approximation as the half of the effective elastic modulus multiplied by the root mean square slope of the surface profile. If this pressure is greater than the hardness of the material and thus
,
The microroughness are completely in the plastic state. For the surface behaves elastically during contact. The size was introduced by Greenwood and Williamson and plasticity index is called. The fact whether the system behaves elastically or plastically, does not depend on the applied normal force.
Method of dimension reduction
Many contact problems can be easily solved using the method of dimensional reduction. In this method, the original three -dimensional system is replaced by a contact with an elastic or viscoelastic Winkler 's subsoil (see image ). The macroscopic contact properties shall coordinate exactly with those of the original system agree, provided that the parameters of Winkler 's bedding and the shape of the body are elected according to the rules of the method.